blind separation for audio signals - are we there yet?

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render some applications if not impossible, at least impractical. Besides audio, an extremely fruitful application arena is digital communications. While the basic ...
Proc. Workshop on Independent Component Analysis and Blind Signal Separation, Jan 11-15, 1999, Aussois, France

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BLIND SEPARATION FOR AUDIO SIGNALS - ARE WE THERE YET? Kari Torkkola Motorola, Phoenix Corporate Research Labs 2100 East Elliot Road, MD EL508, Tempe, AZ 85284, USA [email protected]

ABSTRACT

We attempt to give an overview of current research in blind separation of convolutive mixing of signals, concentrating on audio signals, and methods applicable thereof. We briefly enumerate some application areas, we present two possible taxonomies of separation methods, one based on system parametrizations, the other on different criteria to solve the problem. We wade through the literature following these taxonomies. We also discuss what might yet be missing in the current research.

1. INTRODUCTION We presuppose familiarity with the concepts of Independent Component Analysis (ICA) and Blind Signal Separation (BSS). If this is not the case the reader is advised to refer to [1, 8, 15, 20, 37]. BSS seemingly has a large number of potential applications in the audio realm. The generic application is, of course, separation of simultaneous audio sources in reverberating or echoing environment, that is, in a natural environment, for example, inside a room. We will enumerate here only a few actual applications, the reader is free to use her imagination to develop more. A very desirable application area would be signal enhancement by removing noise or other unwanted signal components using blind separation methods as in [31], for example. In this area only one signal is of interest, the rest is considered as nuisance. Enhancement of voice quality in mobile phones, would be one important application, especially in car environments. Since voice coders used in cell phones are optimized for coding speech alone, the combination of excessive noise with a speech signal results in a poor sound quality. Some initial experiments in this area can be found in [78]. Making voice dialling or speech recognition in general more viable in noisy environments would fall into the same category [115, 48]. Spying, intelligence, or forensic applications fall also under the same category whereby the interest might be in picking up one important signal amongst others [69]. In audio communications transparency refers to reproduced audio being ideally free from reverberation, noise, acoustical echoes, and mixed other speakers [82]. Teleconferencing and speakerphones are two areas where speech signal aquisition with transparency is desirable. Combining existing multi-channel acoustical echo cancellation technology with BSS has been shown to be useful in a teleconferencing setup [82]. Hearing aids are also another lucrative application area for speech enhancement through BSS. Whether some of the above can yet be a viable and profitabe application area, is an open question and will be touched upon in the concluding section. Current limitations in the methods might render some applications if not impossible, at least impractical. Besides audio, an extremely fruitful application arena is digital communications. While the basic concept remains the same (mul-

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tiple transmitters at same frequency, multiple antennas receiving multiple mixtures, for example), there are a few important differences. Signals in this context are man-made and thus their properties are completely known in advance. This can (and should be) be exploited in devising separation methods. Another difference is that signals could be transmitted in short bursts, which might call for block-based algebraic methods rather than adaptive methods [102]. The paper at hand, however, concentrates in methods applicable to audio. The main contribution of the paper is an attempt to collect a major part of the literature pertinent to BSS and audio, and to present that literature in light of two possible taxonomies of separation methods. These are based on how to parametrize the separating structure, and on the criteria used for separation. Due to the breadth of the literature and page limitations, the presentation cannot be but superficial. We also discuss what the technology yet might lack to produce successful applications.

2. PARAMETRIZATION Any method for separation of convolutive mixtures can be roughly divided in three essential components: 1) parametrization of the separation system (filters or matrices), 2) the separation criterion, and 3) the method to optimize the chosen criterion. We concentrate in looking at the first two components, and mention only briefly that the optimization methods can be coarsely divided into adaptive and algebraic approaches, and the former category can further be subdivided into stochastic gradient type algorithms (with or without 2nd order information, i.e. Newton's method), and function zero search algorithms or fixed point methods. The latter category mainly consists of methods to jointly and/or approximately diagonalize a number of matrices. In this section we discuss what alternatives exist for the parametrization of the separating system and possibly for the parametrization of the source signals if the method at hand requires this.

2.1. Feedforward, Feedback? Most real convolutive mixing scenarios with audio can be modeled as a feedforward mixing network having FIR filters in its branches. A room with multiple simultaneous sound sources and multiple microphones is an example, where the mixing filters are room impulse responses between each source and each microphone. The separation system, ideally inverting the effect of the mixing system, can also be modeled as a feedforward network of FIRfilters that approximate the required inverse filters. Consider a simple 2x2 mixing case in the z-domain:

X1 (z) X2 (z)

= =

A11 (z)S1 (z) + A12 (z)S2 (z) A21 (z)S1 (z) + A22 (z)S2 (z);

(1)

Proc. Workshop on Independent Component Analysis and Blind Signal Separation, Jan 11-15, 1999, Aussois, France Solving for the sources gives the ideal solution:

S1 (z) = ( A22 (z)X1 (z) ?A21 (z)X2 (z))=G(z) (2) S2 (z) = (? A12 (z)X1 (z) +A11 (z)X2 (z))=G(z) where G(z ) = A12 (z )A21 (z ) ? A11 (z )A22 (z ). In a feedforward filter separation system with Wij operating on Xj , for example, W11 (z) = A22 (z)=G(z) to reproduce the exact source. How-

ever, the convolutive separation problem has the undeterminacy upto arbitrary filtering of the sources. Depending on the learning algorithm, the sources will be distorted by filtering. Some separation methods tend to produce temporally whitened outputs as they aim at redundancy reduction. In the feedforward case it would be actually very desirable to learn filters without the G(z )?1 factor as the filters to be learned are the same mixing filters and not their inverses (the adjunct of the mixing, that is). There is not, though, an effective way of achieving this. Another possible arrangement is the feedback architecture, which in the 2x2 case is

U1 (z) U2 (z)

= =

X1 (z) ? W12 (z)U2 (z) X2 (z) ? W21 (z)U1 (z);

W12 and W21 have the following ideal separation solution: W12 (z) = A12 (z)A22 (z)?1 W21 (z) = A21 (z)A11 (z)?1 :

(3)

(4)

Note that the inverses of the direct mixing filters are required. If these are minimum-phase filters they have stable causal inverses. That is, if the direct paths are “good” a feedback network with causal filters is able to invert the mixing. Otherwise one must use the feedforward network with acausal filters or implement acausal filters within the feedback network as described in [35].

2.2. Frequency domain The parameters to be learned above are the time-domain coefficients of the filters. However, in the audio case the filters may need to be thousands of taps long to properly invert the mixing. Computationally it may be lighter to move to the frequency domain as convolutions with long filters in the time domain become efficient multiplications in the frequency domain under certain conditions [65, 72]. Now there are two avenues to take. In the first one everything including the actual separation could be done in the frequency domain. This has the great advantage of decomposing a convolutive mixing problem into multiple instantaneous mixing problems (i.e., ICA), that can be solved using any desired method. The downside is that now the standard ICA indeterminacy of scaling and permutation appears at each output frequency bin! Reconstruction of the time-domain output signal requires all frequency components of the same source. Various methods to overcome the scaling and permutation problem using different continuity criteria are presented in [92] and [13, 86, 85, 91, 65, 70, 72, 110]. The second avenue is that the actual separation is not done in the frequency domain but only one or some aspects of the separation algorithm. The rest is done in the time domain. Filters may be easier to lern in the frequency domain as components are now orthogonal and not dependent on each other like the time domain coefficients [7, 81]. Examples of methods that apply their separation criterion (independence, HOS, nonlinearities) in the time domain but do the rest in the frequency domain are reported in [7, 42, 48]. A frequency domain representation of the filters is learned, and they are also applied in the frequency domain. The final timedomain result is reconstructed using standard, e.g., overlap-save signal processing techniques. Thus, the permutation and scaling

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problem does not exist. An example of learning a filter of matrices in time domain, when the criterion is in frequency domain is presented in [58]. Back et al. also present an example of the HJ-algorithm in the frequency domain, whereas the nonlinear functions are applied in the time domain [7]. Another type of parametrization is presented in [93]. The source location is parametrized, and frequency bin information is clustered to produce consistent source location estimates. Alternative discrete time operators are considered in [6].

2.3. Decomposition Rather than trying to learn these possibly huge filters all at once, it is possible to decompose the problem [113, 68]. At this point it is useful to discuss the relation between independent component analysis (ICA) and blind separation of convolutive mixtures (BSCM). ICA (in the separation context) makes use of spatial statistics of the mixture signals to learn a spatial separation system. For stationary sources, ICA needs to use higher than 2nd order spatial statistics to succeed. However, if the sources can be assumed to be nonstationary 2nd order spatio-temporal statistics is sufficient as shown in [97]. In contrast, BSCM needs to make use of spatio-temporal statistics of the mixture signals to learn a spatio-temporal separation system. Stationarity of the sources is decisive for BSCM, too. If the sources are not stationary, only 2nd order spatio-temporal statistics is enough as briefly discussed in [105] and later, for example, in [70, 65]. Stationary sources require again higher than 2nd order statistics, but in the following fashion. Spatio-temporal 2nd order statistics can be made use of to decorrelate the mixtures. This step returns the problem to that of conventional ICA, which again requires higher-order spatial statistics. Examples of these approaches are given in [39, 23, 34] with linear prediction based methods, in [18] with an adaptive approach, and in [52] with a beamforming approach. Alternatively, for sources that cannot be assumed nonstationary one can resort to higher-order spatio-temporal statistics from the beginning as done in a lot of papers. Another way to decompose the problem is presented in [67]. In this case the the microphone arrangement is rather peculiar, a compact microphone array: two omnidirectional microphones 1cm apart in a reverberant environment. Now, for each source, the transfer functions differ predominantly by a small delay. Mixtures are modeled as x = DRs, where D is a matrix of bandlimited approximations of delta functions at some delays, and R represents ^ all other acoustic effects. First stage estimates delays D

y = adj (D^ )x = adj (D^ )DRs = Hs (5) Second stage cancels off-diagonal elements of H (small relative to diagonal elements) using a feedback configuration

z = y ? Mz = (M + I )?1 Hs: (6) Now H is easier to estimate than DR directly since its offdiagonal elements are small: Initial guess M = 0 is likely to lie near a global optimum. Simple decorrelation adaptation is then sufficient to learn M .

2.4. Sources parametrized, too Some separation methods require assumptions about the sources (see Sec. 3.3) and parametrize some aspects of them, such as the densities, some parameters thereof [64, 31, 5], or some parameters related to temporal statistics of the sources, such ar AR parameters [73, 71, 49].

Proc. Workshop on Independent Component Analysis and Blind Signal Separation, Jan 11-15, 1999, Aussois, France

2.5. System identi cation approach With room acoustics the inverse system generally contains more parameters than the mixing system. This would suggest rather learning the mixing system than the separation system.

3. SEPARATION CRITERIA The actual separation criterion can be quite independent of the chosen parametrization. In this section we go through a major part of them that have appeared in the literature in the audio context.

3.1. Minimizing mutual information

Y

Independence criterion can be based on factoring the marginal output density. pu( ) = pui (ui ): (7)

u

i

Divergence between the two sides of (7) reflects thus the independence of the ui . For the two sides of (7) the divergence is

D(pu (u);

Yp i

ui (ui ))

= =

Z

pu (u)log Q ppu(u()u ) du

?H (U ; W ) +

i ui i

(8)

X H (U ; W ); i

i

the joint entropy of output minus the sum of entropies of individual components, which is also the mutual information between the output components. Minimizing this expression results in statistical independence of the output components. If W is just a matrix, and u = Wx then

H (U ; W ) = H (X ) + log det(W )

(9)

Minimizing this mutual information requires entropies of individual components, which are not available, but can be estimated by means of statistical expansions [20, 1, 36] or other methods [38]. In the convolutive mixture context MMI has been used in [58, 19, 17].

3.2. InfoMax, or Entropy maximization Consider now passing the outputs through bounded nonlinear functions, that approximate the cumulative densities of the sources y = g(Wx). If outputs u are the separated true sources, the y have density close to a uniform density, the density that has the largest entropy among bounded distributions. Maximizing the entropy of y is thus equivalent of producing the true sources. This criterion makes use of the source densities implicitly. This criterion was applied to delayed and convolutive mixing in [99, 98] and further developed in [19, 47, 112, 2, 3, 27, 35]. Fully or partially frequency domain approaches were developed in [49, 48, 41, 90, 89, 91].

3.3. Latent source models and ML Source densities can be made use of explicitly, too. If they are known, or if their shapes are known less perhaps some parameters that need to be estimated while is learned, the divergence between the marginal output density and the known source densities can be used as the criterion to be minimized, that is, find that drives the outputs towards known distributions. This approach lends itself naturally to the maximum likelihood estimation [75, 9, 14]. With convolutive mixing this approach has been taken in [71, 73, 64, 28, 31].

W

W

3

Even if the densities are unknown we can estimate them as sums of some simple, for example Gaussian or logistic distributions while the separation matrix is being estimated. Either gradient ascent [74] or expectation-maximization in some cases [10, 64] can then be used. This situation lends itself also to the following interpretation. The sources can be interpreted as latent signals whose convolutive mixture is the only observable [5, 4]. General literature on latent models and their estimation is then applicable.

3.4. Bussgang approach Bussgang methods have been used as tools for blind deconvolution, where the true source is estimated from a signal corrupted by convolutional noise by a nonlinear function which optimally equals g (s) = (@ps (s)=@s)=ps (s). Equalization algorithms are derived by finding a filter that minimizes the difference between the output and the true source estimated through g . LMS is typically the criterion. Extensions for multichannel blind deconvolution and separation were presented by Lambert [42, 43, 45]. Coupled with FIR-matrix algebra [46, 43] efficient separation methods seem to result. See also [106, 107] for application of these methods for the overdetermined mixing case. It is notable that the nonlinearity has the same exact form as in the entropy maximization, and maximum likelihood approaches, leading to similar separation algorithms [50].

3.5. Central Limit Theorem Assumption of non-Gaussian sources enables us to apply the central limit theorem: The sum of a “small” number of non-Gaussian signals has a distribution that is “closer” to a Gaussian than the densities of the sources. The distance of the marginal density of the output pu to a Gaussian can be expressed by using the divergence, which is equivalent to the differential entropy of a Gaussian (which has the same mean and covariance as pu ) minus the differential entropy of pu . This measure is also called negentropy. From this measure it is straightforward to derive a stochastic gradient ascent adaptation rule to find a that maximizes this distance and thus separates the sources, as shown by Girolami in [33]. Deviation from Gaussianity can also be measured directly by higher order statistics since these are zero for Gaussians. For example, kurtosis, which is written as E u4 3(E u2 )2 for a zero mean variable, could be used. Both kurtosis and negentropy require us to know to which direction we are driving the output densities from a Gaussian. For example, densities that have negative kurtosis (flat densities like uniform distribution) require the kurtosis of the output signal to be minimized whereas for positively kurtotic (sharply peaked) signals kurtosis needs to be maximized. Either we need to know the signs of kurtoses of our sources, or we as discussed in [33]. need to estimate them while learning the

W

f g?

f g

W

3.6. Minimization of cross-HOS As mentioned in Section 2.3 it is possible use the traditional ICAmethods by constructing the criteria for separation from high-order statistics, but by doing it spatio-temporally. One can construct an objective function, usually from fourth order cross-cumulants, and minimize this using stochastic gradient descent on the filters [30, 66, 114, 22, 100], or a contrast function as in [63]. Alternatively, one can construct a simple algorithm that aims just at canceling the criteria [94]. HOS can also be used implicitly through nonlinear functions. Examples and analysis are given in [95, 40, 94, 25, 7, 57, 24]. Platt and Faggin first derived similar rules employing nonlinear functions, but from the minimum output power principle [76]. Approaches that utilize HOS in the frequency domain are

Proc. Workshop on Independent Component Analysis and Blind Signal Separation, Jan 11-15, 1999, Aussois, France described in [13, 84, 86, 87]. An algebraic approach is presented in [96]. In general, estimation of higher-order statistics is more sensitive to noise and outliers than that of 2nd order statistics. Thus, based on this argument it would seem to be more robust to work on 2nd order statistics as much as possible.

3.7. Spatio-Temporal decorrelation Let us look at spatial covariance in instantaneous separation

Rx (t) = ARs (t)AT + Rnoise (t); x(t) = As(t) + n(t) (10) If s is non-stationary, that is, Rs (t) 6= Rs (t +  ) we get multiple conditions for different choices of  to solve for A, Rs (t), and Rnoise (t), where the covariances are diagonal matrices. For convolutive mixtures we may look at cross covariances over time Rx (t; t +  ) = E fx(t)x(t +  )T g. This approach was mentioned in [105] and utilized in [61] and [59]. In frequency domain for sample averages we can write

R x (!; t) = A(!)Rs (!; t)AH (!) + Rnoise (!; t)

(11)

Again, if s is non-stationary we can write multiple linearly independent equations for different time lags and solve for unknowns or find LMS estimates of them by diagonalizing a number of matrices in frequency domain [29, 108, 65, 70, 72, 60, 110]. For minimum-phase mixing decorrelation only can provide a unique solution without having to make use of the non-stationarity [12, 55, 54, 88]. There is a multitude of adaptive approaches [101, 103, 104, 105, 57, 55, 16, 115, 26, 78, 79, 81, 82], a few algebraic [56], and many that are derived from anti-Hebbian learning rule considerations [59, 77, 32, 18, 31].

4. DISCUSSION Despite the seeming scope of research good results in realistic scenarios are hard to come by. What makes it difficult to move BSS into real world applications? Are there some inherent limitations in the audio setup that make it an extremely hard (if indeed solvable) problem? We pose some questions and try to present some directions in this section. Are all assumptions really true? Some methods assume stationarity of the sources, some do not. Speech is certainly nonstationary in larger time scale, but quasi-stationary in a short scale. Derivation of some methods might expect doubly-infinite filters, which can only be approximated in reality. This approximation might cause local minima in optimizing the chosen criterion [88] Are there always (much) more sources than sensors? An intriguing application would be to separate speech from car noise in mobile communications. However, one quickly realizes that there are multiple noise sources, in fact an infinite number, since the whole car interior is vibrating and acts as a delocalized noise source. A 2x2 speech-noise separation system would not work in this case. Is the reality too dynamic? In general, the reported performance figures for static sources (loudspeakers) are higher than figures for real people, even in seemingly static positions. The effect of a speaker turning her head 10-20 degrees, or leaning backwards a couple of inches can have a drastic effect to the impulse response between the speaker and the microphones. These effects have been studied in acoustics, and further cross-fertilization between the fields would seem to be necessary to establish bounds to what performance could be expected from BSS in situations involving live speakers. Bradstein simulates these effects in [11] and concludes that “Any system which attempts to estimate the reverberation effects and apply some means of inverse filtering would

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have to be adaptable on almost a frame-by-frame basis to be effective.” This gives quite a pessimistic view for applications that involve on-line adaptation to dynamic situations. Related to the previous question is how much data do we really need to converge? Are there too many parameters? Filters with thousands of taps in the time domain need tens (or hundreds) of thousands of samples to converge. That might be too much for the application. What else can be done? It would be reasonable to combine different approaches to make use of every bit of available knowledge. For example, we should combine the nonstationarity-exploiting decorrelation approaches with approaches that make use of the source densities. Initial attempts are reported in [51]. A promising approach seems to be decomposing the problem into smaller and independent subproblems instead of trying to solve all at once whith a huge number of parameters. The sensitivity to (unavoidable) noise is much examined in communications but has not been studied enough in the audio context. Some studies or methods taking noise into account are presented in [21, 62, 64, 72]. Using more microphones than there are sources to separate should in theory be able to improve the noise tolerance, in addition to separation quality. Westner, though, reports rather pessimistic results in his initial experiments [106]. An avenue that has not been looked at enough in BSS is to make use of the fact that the target signal is a speech signal. In [11] Brandstein advocates explicitly incorporating the nature of the speech signal including non-stationarity, model of production, pitch, voicing, formant structure, and a model of source radiation, into a beamforming context. He feels that this is essential to realize the goal, high-quality speech signal acquisition from an unconstrained talker in a hands-free environment surrounded by interfering sources. This goal equals the goal of many BSS applications. Some recent BSS work to these directions is presented in [111, 109]. The last but the not the least question is What is the best method for separating audio signals mixed convolutively? Given the breadth of the field and the lack of common criteria and databases this question remains unanswerable. It is possible to derive some theoretical bounds for some algorithms [80, 53, 113], but not for all of them. Empirical comparisons using agreed upon databases and measurements may be the only way to find partial answers [83, 44]. We conclude with these open questions, and state that as coping with artificial signals seems to be more straightforward than with natural signals, it might be that BSS with convolutive mixtures finds it first real applications in the area of communications.

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